The paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry. In particular, it contains an axiomatic treatment of Gromov–Witten classes, and a discussion of their properties for Fano varieties. Cohomological Field Theories are defined, and it is proved that tree level theories are determined by their correlation functions. Application to counting rational curves on del Pezzo surfaces and projective spaces are given. Let V be a projective algebraic manifold. Methods of quantum field theory recently led to a prediction of some numerical characteristics of the space of algebraic curves in V, especially of genus zero, eventually endowed with a parametrization and marked points. It turned out that

Mirror Symmetry was discovered several years ago in string theory as a duality between families of 3-dimensional Calabi-Yau manifolds (more precisely, complex algebraic manifolds possessing holomorphic volume elements without zeroes). The name comes from the symmetry among Hodge numbers. For dual Calabi-Yau manifolds V, W of dimension n (not necessarily equal to 3) one has dim H p (V, Ω q) = dim H n−p (W, Ω q). Physicists conjectured that conformal field theories associated with mirror varieties are equivalent. Mathematically, MS is considered now as a relation between numbers of rational curves on such a manifold and Taylor coefficients of periods of Hodge structures considered as functions on the moduli space of complex structures on a mirror manifold. Recently it has been realized that one can make predictions for numbers of curves of positive genera and also on Calabi-Yau manifolds of arbitrary dimensions. We will not describe here the complicated history of the subject and will not mention many beautiful contsructions, examples and conjectures motivated

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The goal of this paper is to give in outline a new proof of the fact that the Floer cohomology groups of the loop space of a semi-positive symplectic manifold (M;!) are naturally isomorphic to the ordinary cohomology of M . We shall then outline a proof that this isomorphism intertwines the quantum cup-product structure on the cohomology of M with the pair-ofpants product on Floer-homology. One of the key technical ingredients of the proof is a gluing theorem for J-holomorphic curves proved in [20]. In this paper we shall only sketch the proofs. Full details of the analysis will appear elsewhere. 1 Introduction The Floer homology groups of a symplectic manifold (M;!) can intuitively be described as the middle dimensional homology groups of the loop space. The boundary loops of J-holomorphic discs in the symplectic manifold with center in a given homology class ff 2 H (M) (integral homology modulo torsion) form a submanifold of the loop space of roughly half dimension and should ther...

Introduction. There has been a recent explosion in the number of mathematical publications due to the discovery of a certain duality between some families of Calabi-Yau threefolds made by a group of theoretical physicists (see [11,26] for references). Roughly speaking this duality, called mirror symmetry, pairs two families F and F ∗ of Calabi-Yau threefolds in such a way that the following properties are satisfied:

Since Gromov introduced his pseudo-holomorphic curve theory in 80’s, pseudo-holomorphic curve soon became a predominant technique in symplectic topology. Many important theorems in symplectic topology have been proved by the technique of pseudo-holomorphic curve, for example, the squeezing theorem [Gr], the rigidity [E], the classifications of rational and ruled symplectic 4-

Contents 0. Introduction 1 1. Stable maps and their moduli spaces 10 2. Boundedness and a quotient approach 12 5. The construction of M g;n (X; fi) 25 6. The boundary of M 0;n (X; fi) 29 7. Gromov-Witten invariants 31 8. Quantum cohomology 34 9. Applications to enumerative geometry 38 10. Variations 43 References 46 0. Introduction 0.1. Overview. The aim of these notes is to describe an exciting chapter in the recent development of quantum cohomology. Guided by ideas from physics (see [W]), a remarkable structure on the solutions of certain rational enumerative geometry problems has been found: the solutions are coefficients in the multiplication table of a quantum cohomology ring. Associativity of the ring yields non-trivial relations among the enumerative solutions. In many cases, these relations suffice to solve the enumerative problem. For example, let N d be the number of degree d, rational plane curves passing through 3d \Gamma 1 general points in P . Since there is a un

We describe an isomorphism of categories conjectured by Kontsevich. If M and ˜ M are mirror pairs then the conjectural equivalence is between the derived category of coherent sheaves on M and a suitable version of Fukaya’s category of Lagrangian submanifolds on ˜ M. We prove this equivalence when M is an elliptic curve and ˜ M is its dual curve, exhibiting the dictionary in detail.